Modes

Top: d'Alembert's picture, two half-pulses travelling at ±v and reflecting with inverted polarity at the fixed ends. Second panel: the separated solution truncated to N modes (black) against the exact motion (dashed). With few modes the sum is a smooth blur; as N grows it sharpens into the two travelling pulses. Standing waves and travelling waves are the same solution written in two bases. The bars show the modal weights sin(nπx_s/L)F(ωₙ): plucking at L/2 removes every even mode, at L/7 the 7th and its multiples, which is why piano hammers strike near 1/7–1/9 of the string. A wider pulse (larger τ) lowers F at high ωₙ: fewer modes, mellower sound.

The equation inside is the same in every case: X″ + k²X = 0. Only the boundary conditions change, and they alone select the allowed k_n. For the pipes the blue curve is displacement and the orange one is pressure: an open end is a pressure node and displacement antinode, a closed end the reverse, so the open–closed pipe keeps only odd harmonics (1, 3, 5, …). The spring-mounted end imposes T y′(L) = −K y(L), the mixed condition c₁y + c₂y′ = 0 of a Sturm–Liouville problem. Its eigenvalues solve tan(kL) = −kL/β: as the stiffness grows, every one slides continuously from (n − ½)π (free end) to nπ (fixed end), and in between the overtones are no longer integer multiples of the fundamental.

The radial factor plays the role of sin(kx) on the string, but J_m is not periodic: the gaps between its zeros start larger than π and approach π only far from the centre (right panel), so the overtones fall at 1.594, 2.136, 2.296, … times the fundamental instead of 2, 3, 4. That inharmonic ladder is why a drum has no clear pitch. Striking the centre excites only m = 0 modes; striking off-centre brings in the others. The same separation in spherical coordinates gives the free oscillations of the Earth, with spherical Bessel functions in r and spherical harmonics in (θ, φ).